Lookahead Saturation with Restriction for SAT

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1 Lookahead Saturation with Restriction for SAT Anbulagan 1 and John Slaney 1,2 1 Logic and Computation Program, National ICT Australia Ltd., Canberra, Australia 2 Computer Sciences Laboratory, Australian National University, Canberra, Australia {anbulagan, john.slaney}@nicta.com.au Abstract. We present a new and more efficient heuristic by restricting lookahead saturation (LAS) with NVO (neighbourhood variable ordering) and DEW (dynamic equality weighting). We report on the integration of this heuristic in Satz, a high-performance SAT solver, showing empirically that it significantly improves the performance on an extensive range of benchmark problems that exhibit hard structure. 1 Introduction During the last decade, many new techniques have been proposed to enhance the performance of the DPLL procedure for solving various hard real-world problems represented in conjunctive normal form (CNF). One of the main improvements of this decision procedure has been the development of better branching variable selection through the use of unit propagation (UP) heuristics [1], which detect failed literals through a one-step lookahead. The effect of integrating the UP heuristic into DPLL is to prune the search tree earlier. In this paper, we provide a new heuristic, DEW-NVO-LAS, which restricts lookahead saturation (LAS) with NVO (neighbourhood variable ordering) and DEW (dynamic equality weighting). DEW weighs equality literals during NVOrestricted lookahead saturation, firstly to restrict the variables to be propagated through the lookahead process, and secondly so that the next branching variable chosen can be the one having the highest score. We report on the integration of the DEW-NVO-LAS heuristic into Satz, showing empirically that it significantly improves Satz s performance on a range of benchmark problems, such as bounded model checking, cryptographic key search, FPGA routing, equivalence checking in circuits, and, particularly, the challenging 32-bit parity learning problems. The same problems are used for a comparative study between Dew Satz, the DEW-NVO-LAS-enhanced Satz solver, and other state-of-the-art SAT solvers. 2 Lookahead Saturation with Restriction Lookahead saturation (LAS) based DPLL was studied in [2]. The key idea underlying LAS is to choose a branching variable which is really the best from an irreducible sub-formula at a given node of search tree. LAS is very similar to the singleton arc consistency (SAC) algorithm in CSP reasoning [3].

2 Intuitively, although a reasoning-intensive process such as LAS can reduce the search tree size enormously, this increased efficiency is outweighed by the cost in terms of runtime. For that reason, we restrict the LAS process using the NVO and DEW heuristics. While the NVO heuristic concentrates on restricting the number of variables to be examined in the next iterative lookahead process by considering only the neighbours of the currently assigned variable in the currently size-reduced clauses, the DEW heuristic restricts the number of literals to be examined during the iterative lookahead process. DEW alone is not particularly useful in this regard, it must be incorporated into NVO-LAS to be really effective. The basic concept of DEW is as follow. Whenever the binary equality clause x i x j, which is equivalent to 2 CNF clauses x i x j and x i x j, occurs in the formula at a node, Satz needs to perform the lookahead process on x i, x i, x j, and x j. As result, variables x i and x j will be associated the same weight, (i.e. 3 following the computation at line 25 of Algorithm 1). Clearly, the processing of x j and x j is redundant, so avoid it by assigning the implied literal x j (x j s) the weight of its parent literal x i (x i s), and then by restricting the lookahead process to literals with weight zero. By doing so, we save two lookahead processes. To clarify the concept, we present a concrete example. Consider the following simple formula with binary equality clauses: (x 1 x 2 ) (x 2 x 3 ) (x 1 x 4 ). The Satz solver evaluates iteratively each variable of the formula by two forced unit propagations, where there is no failed literal found. Each literal of the formula gets the same weight, i.e. 3. Intuitively, we do not need to lookahead on variables x 2, x 3 and x 4 after performing lookahead on x 1 : all three get the weight of the parent x 1. The effect of the DEW heuristic is that the weight of each implied literal accumulates dynamically during the lookahead process, and if it is greater than zero then no lookahead process is done on that literal. The DEW heuristic is executed only whenever binary equality clauses occur in the current state formula. Our main observation is that DEW benefits markedly from NVO-LAS. We integrate DEW-NVO-LAS heuristic into Satz, and call the new solver by Dew Satz. Intuitively, the merged heuristic will enhance the performance of NVO-LAS by avoiding the redundant lookahead process, which is computed by DEW. At the same time the DEW heuristic benefits from NVO-LAS as this dynamically bounds the number of variables to be weighed. The two mutually compatible heuristics work together to improve lookahead-based DPLL. Dew Satz also inherits from Satz a preprocessor for saturating the input clauses under resolution with the restriction to clauses of length 3, removing subsumed clauses and tautologies along the way. In certain cases, the preprocessor may remove some equality clauses. Algorithm 1 sketches the branching rule of Dew Satz. The procedure Compute DEW(x i ) is called for weighting the implied literals of the parent variable x i. The function UP(F i ) at line 7 (10) of Algorithm 1 is executed if w(x i )=0 (w( x i )=0). When there is no conflict found during the two unit propagations, then variable x i will be piled into the branching variable candidates stack B.

3 Algorithm 1 DEW-NVO-LAS-BranchingRule(F) 1: Push each variable x i V to NVO STACK; 2: repeat 3: B := ; F init := F; 4: for each variable x i NVO STACK do 5: Let F i and F i be two copies of F; 6: if w{x i} = 0 then 7: F i := UP(F i {x i}); 8: end if 9: if w{ x i} = 0 then 10: F i := UP(F i { x i}); 11: end if 12: if empty clause F i and empty clause F i then 13: return unsatisfiable ; 14: else if empty clause F i then 15: F := F i ; NVO(x i); 16: else if empty clause F i then 17: F := F i; NVO(x i); 18: else 19: w(x i) := diff (F i, F); w( x i) := diff (F i, F); 20: B := B {x i}; Compute DEW(x i); 21: end if 22: end for 23: until F = F init 24: for each variable x i B do 25: W(x i) := w(x i) w( x i) + w(x i) + w( x i); 26: end for 27: NVO(x i); 28: return x i with highest W(x i) to branch on; 3 Experimental Results The 32-bit parity problem instances are considered as a challenging problem [4]. To answer the challenge, equality reasoning has been integrated differently in different solvers [5 8]. EqSatz uses equality reasoning in the search process while Lsat and March eq use it in their preprocessors. In Table 1, we present the performance of Dew Satz on par16* and the challenging par32* instances in comparison with the following state-of-the-art solvers: EqSatz, Satz (ver. Satz215), zchaff (ver ), March eq (ver. March eq 010), Lsat (ver. 1.1). It is important to observe that Dew Satz can solve the 32-bit parity problem in the range of 411 to 17,564 seconds. It solved the par32-5 and par32-5-c instances without using the preprocessor (with preprocessing, these instances took 27 and 29 hours respectively). The results of Dew Satz refute the pessimistic view that lookahead-based DPLL must perform poorly on such highly structured problems. In order to evaluate further the performance of Dew Satz versus other solvers used above, we extended the empirical study to include some well-known circuit-

4 Instance (#Vars/#Cls) Satz Dew Satz EqSatz Lsat March eq zchaff par16* par32-1 (3176/10227) >24h 12, >24h par32-2 (3176/10253) >24h 5, >24h par32-3 (3176/10297) >24h 7,198 2, >24h par32-4 (3176/10313) >24h 11, >24h par32-5 (3176/10325) >24h 17,564 2, >24h par32-1-c (1315/5254) >24h 10, >24h par32-2-c (1303/5206) >24h >24h par32-3-c (1325/5294) >24h 4,474 1, >24h par32-4-c (1333/5326) >24h 7, >24h par32-5-c (1339/5350) >24h 11,899 2, >24h Table 1. CPU time (in seconds) comparison. >24h shows that the problem cannot be solved in 24 hours. Problem Dew Satz Satz EqSatz March eq zchaff barrel barrel , barrel8 72 >5, barrel9 158 >5, longmult longmult ,625 longmult , ,643 longmult , ,225 longmult , ,456 longmult , queueinvar queueinvar queueinvar queueinvar queueinvar cnf-r3*(8) (1) 12,032 2, bart*(21) (17) 85,403 (1) 6, homer*(15) 3,054 (15) 75,000 (15) 75,000 (15) 75,000 (1) 9,245 lisa*(14) 2,955 1,721 5,788 1,211 (3) 30,349 hwb-n20*(3) ,355 hwb-n22*(3) ,700 hwb-n24*(3) 3,457 3,115 4,170 1,115 (3) 15,000 pb-sat*(12) (2) 13,818 (2) 15,793 (4) 24,478 7,869 (4) 21,099 pb-unsat*(12) 19,547 22,269 (4) 24,293 (1) 19,659 (8) 44,144 philips-org 697 3, >5,000 >5,000 philips 295 1, >5,000 Table 2. CPU time (in seconds) on realistic benchmark problems.

5 related benchmark problems. All problems used in the study are taken from SATLIB ( where some of them are used in previous SAT competitions. Some of the problem instances contain more than 600,000 variables. The problems cnf-r3*, bart*, lisa* and pb-sat* are satisfiable, and the others are unsatisfiable. The timebound for this experiment is 5,000 seconds per problem instance. Table 2 shows the runtimes of Dew Satz, Satz, EqSatz, March eq and zchaff on these problems. The numbers of instances of some problems are indicated in brackets after the problem names, and the number of instances on which each solver failed is also indicated in brackets before the total time. We count 5,000 as the increment in runtime for an unsolved instance. The experimentations were conducted on Intel Pentium 4 PCs with 3 GHz CPU, under Linux. In general, this extended study further confirms the superior performance of Dew Satz in comparison with the other four solvers. Where Dew Satz fails to solve 2 instances in the given timebound (instance pb-sat needs 6,005 seconds and instance pb-sat needs 12,958 seconds), Satz, EqSatz, March eq and zchaff fail on 20, 40, 18 and 21 instances respectively of the 108 given. The empirical results also show that unit propagation based lookahead in DPLL is still a powerful technique. Simply enhancing it with a straightforward heuristic allows us to solve many more hard problems, as shown by the results above. Acknowledgments This work was funded by National ICT Australia (NICTA). National ICT Australia is funded through the Australian Government s Backing Australia s Ability initiative, in part through the Australian Research Council. References 1. Li, C.M., Anbulagan: Heuristics based on unit propagation for satisfiability problems. In: Proceedings of 15th IJCAI, Nagoya, Aichi, Japan (1997) Anbulagan: Extending unit propagation look-ahead of DPLL procedure. In: Procs of 8th PRICAI, Auckland, New Zealand, Springer, LNAI 3157 (2004) Bessière, C., Debruyne, R.: Theoretical analysis of singleton arc consistency. In: ECAI-04 Workshop on Modeling and Solving Problems with Constraints, Valencia, Spain (2004) Selman, B., Kautz, H., McAllester, D.: Ten challenges in propositional reasoning and search. In: Proceedings of 15th IJCAI, Nagoya, Aichi, Japan (1997) Warners, J.P., van Maaren, H.: A two-phase algorithm for solving a class of hard satisfiability problems. Operations Research Letters 23 (1998) Li, C.M.: Integrating equivalency reasoning into Davis-Putnam procedure. In: Proceedings of 17th AAAI, USA, AAAI Press (2000) Ostrowski, R., Grégoire, E., Mazure, B., Sais, L.: Recovering and exploiting structural knowledge from CNF formulas. In: Proceedings of 8th CP. (2002) Heule, M., van Maaren, H.: Aligning CNF- and equivalence-reasoning. In: Proceedings of 7th SAT, Vancouver, Canada (2004)